Finding Large H-Colorable Subgraphs in Hereditary Graph Classes

W pracy badamy złożoność problemu znajdowania największego indukowanego podgrafu, który jest homomorficzny z ustalonym grafem H. Pokazujemy, że przy pewnych założeniach na instancje wejściowe, problem może być rozwiązany w czasie wielomianowym (lub podwykładniczym).

We study the Max Partial H-Coloring problem: given a graph G, find the largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without loops. Note that when H is a complete graph on k vertices, the problem reduces to finding the largest induced k-colorable subgraph, which for k = 2 is equivalent (by complementation) to Odd Cycle Transversal. We prove that for every fixed pattern graph H without loops, Max Partial H-Coloring can be solved: - in {P₅,F}-free graphs in polynomial time, whenever F is a threshold graph, - in {P₅,bull}-free graphs in polynomial time, - in P₅-free graphs in time n^O(ω(G)), - in {P₆,1-subdivided claw}-free graphs in time n^O(ω(G)³). Here, n is the number of vertices of the input graph G and ω(G) is the maximum size of a clique in G. Furthermore, by combining the mentioned algorithms for P₅-free and for {P₆,1-subdivided claw}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial H-Coloring in these classes of graphs. Finally, we show that even a restricted variant of Max Partial H-Coloring is NP-hard in the considered subclasses of P₅-free graphs, if we allow loops on H.